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\title{Using sparse description of deformation fields as means to classification}
\author{Nishith Tirpankar, Anshul Joshi, Wathsala Widanagamaachchi}

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\maketitle

%1. sparsity - how it affects the classification
%2. error metrics

\section{Problem}
Handwritten digits classification is a classic machine learning problem. This involves dealing with high dimensional data. Say, if the 
size of an image is 16x16 then the variability in the images is on the dimensionality of 256. Classification on a high dimensional data
is computationally expensive.

Although the variability in images is high dimensional, the inherant variability in an images can be expressed on a much smaller 
dimensionality.

We have reformulated the classification problem to an image registration problem using landmarks. The use of landmarks reduces the
dimensionality. Intuitively a larger dimensionality will give us a very small error in classification. We want to find out how much we 
can reduce the number of landmarks and hence the dimensionality to still get a reasonable classification rate.

\section{Team Menbers}
Nishith Tirpankar, Anshul Joshi, Wathsala Widanagamaachchi

\section{Approach}
Our approach to answering the problem consists of 3 steps. In the first step we will implement landmark matching as described in
[1] for registration of two images. Next, we will extend the pairwise registration to construct the atlas. Please note that the
data we will be using for classification is not the images of the digits but the momenta vectors.
The second step involves classification. We register the test image with the images of each if the classes. The deformation fields of
each such registration $\mathbf{\alpha_c}$ where $c$ denotes the class which is $c\in{0,1,...9}$ are now compared with the deformation
fields in the atlas formed in the earlier step. The comparision criterion we have used is minimum Mahalanobis distance.

The sparsity of the deformation fields vectors $\mathbf{\alpha}$ can be modulated based upon $\gamma_{sp}$. In the third step we will
find how the classification error varies with the sparsity. The sparsity defines the dimensionality of the data used in classification.
The higher the sparsity the lower is the dimensionality of the data. Hence, we wish to find out how the sparsity affects the error rate
of the classifier.

We will be experimenting with other classification methods as well as other methods of measuring error on the testing dataset.

% training step: Refer to the sparse parameterization paper and say that we used that approach to get the atlas based on a training dataset.
% We need to describe the deformation method that give the deformation vectors.
% classification step: if we get an input image then we get try to classify that image by looking at the deformation vectors of the test
% image with the deformation vectors in the atlas. we classify using smallest mahalanobis distance - describe this. We may use some other
% classification criterion later. The test and training dataset contain an equal number of images for each class.
% how to measure error: plot error rate vs gammaSP for various values of prior ie. no of control points. we expect to see a knee curve. 

\section{Dataset}
The dataset selected for our case is the zip digits handwritten database taken from \url{http://www-stat.stanford.edu/~tibs/ElemStatLearn}
made available by the neural network group at AT\&T Research Labs thanks to Yann Le Cunn. It consists of two repositories of training and 
test dataset respectively. The datasets are of normalized handwritten digits automatically scanned from the envelopes by the U.S. Postal 
Service. The original scanned digits are binary and of different sizes and orientations; the images have been deslanted and size 
normalized resulting in 16x16 grayscale images. There are 7291 images in the training dataset and 2007 images in the test dataset.

\section{Timeline}
Pairwise registration - 17 October
\\Atlas generation - 1 November
\\Classification using Mahalanobis distance as a criterion - 5 November
\\Computing error based upon any error metric - 15 November

\section{References}
[1] Stanley Durrleman, Marcel Prastawa, Guido Gerig, and Sarang Joshi. Optimal Data-driven Sparse Parametrization of Diffeomorphisms for 
Population Analysis. Information Processing in Medical Imaging (IPMI) 2011. Lecture Notes in Computer Science (LNCS) 6801, Pages 123-134.

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